
"Integral Mathematics: An AQAL Approach"

03082007, 11:28 AM 

iiadmin



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"Integral Mathematics: An AQAL Approach"


03122007, 12:06 AM 

ralphweidner



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Re: "Integral Mathematics: An AQAL Approach"
what do we mean by integral mathematics? is it the mathematics of perspectives that ken has initiated? ideally, i guess it would be, but in that case ken appears to be the only contributor. elliot benjamin, for example, has wisely refrained, imo, from attempting to add anything significant to what ken has already done.
it's a very long ways from conventional mathematics to a mathematics of perspectives. as well as i can tell, (pure) math works with interior objects. the number 1, for example, is not something i can see out there, resting in the sun, say, with my eye of flesh. i have to use the eye of mind. even in the application of math i must have recourse to the interior objects of pure math, although they might have referents that are exterior objects. for example, the pure math formula 1+2=3, when applied to something, asserts that when i take one of that thing and add it to two of those things, i will get three of those particular things. and that will apply to anything for which it makes sense, i.e. it makes sense to talk about two or three of those things. we could apply the formula to dogs, or to conscious experiences of nonduality, say.
while interior objects take precedence over exterior objects in math, especially pure math, exterior objects can enter not only in applications, but more significantly, in the development of mathematics. for the interior objects of math quite often were first abstracted from exterior objects. the notion of a triangle, say, most probably developed out of seeing things that looked like triangles.
like so many other things, the development of math as a whole depends integrally on the interplay of its two sides, pure and applied. most of the math we know today developed with a view to applications in the RQs. it seems to me that there is a good historical reason for this, at least before the emergence of green. for at an orange altitude, with the ability to take a third perspective, one is well equiped to abstract from exterior objects, but not from interior objects, for which the ability to take a fourth perspective is crucial, as i understand, the added perspective being the 1st person perspective that illuminates our interiors, taking a 2nd perspective, of which, is needed just to see an interior object. please correct me, anyone in the know, if i'm mistaken here, as i very well could be!
anyway, assuming i'm more or less correct, what with all the green mathematicians who must be coming onto the scene, is there any reason other than lack of creativity in present society that they couldn't be looking much more at interior objects as candidates for applications? imagine the entirely new math that could come out of this!
in line with elliot benjamin's article, let me suggest some very elementary examples. he applies, for example, his notion of a biquasigroup in order to suggest a simple model for transformation and development. whereas he is an algebraist, i am a geometer in very general, orienting terms. so i'm not even completely sure i understand him, but he seems to me to be using an algebraic model in place of the geometric model of a spiral that most of us are more familiar with. both have their distinct advantages. in both cases we get this sense of developing over a period of time until we get to a point where in some sense we're where we were before, but in some other sense we're not: we've somehow transformed, moved on to the next wave. we've gone completely around the spiral, but somehow ended up above where we were before. in the case of the biquasicyclic group, we've gone through the complete cycle and come back to the element we began with, but it's now a transformation of that element to a higher level or wave.
a traditional clock conveniently includes versions of both of these models. in this case we begin at some time, say 12 o'clock, and, neglecting the second hand, we follow the minute hand around the clock till we get back to 12 where we began, but not quite, because in the mean time, the hour hand has crept its way onto 1 (but not as in, i crept into the crypt and cried, sob, sob). this version allows for as many as 12 transformations before it selfdestructs. it combines the continuous sense of development found in going up the spiral (the movement of the minute hand) with the discontinuous sense of transformation found in going through the cycle of quasigroup elements (the numbers on the clock). of course, these are extremely simple models, or metaphors, for development and transformation, but they can serve as heuristic devices, especially when this is new to us.
a somewhat more sophisticated model is suggested by ken's description of stages as probability waves. as he has said, someone judged by one of the tests now available to be at an altitude of green, say, along the developmental line corresponding to that test, has in very general, generic terms, answered only about 50% of the test questions as green would, something like 25% as orange would, and 25% as teal would. so the test actually gives us a probability wave, not a point or an element as in the previous models, as granulated as it is capable of distinguishing, at the time the person took the test. if this test is given to a wide population at regular intervals of time, i.e. longitudinally, then it will be possible to (re)construct models of development along that line for various subpopulations, which will show how their probability wave generally moves , or doesn't, up that developmental line with time.
notice that the mathematical models i've been using already exist, so this is not yet new mathematics, just a new application. hopefully, as we begin to understand development and transformation better, someone will be able to abstract out of all the detail what essentially is happening. that could be the beginning of a new mathematics.
there are already attempts to understand better just what distinguishes a stage from the one that succeeds it by means of transformation. kegan has done this for the selfrelated line he's been studying of 'orders of consciousness'. conceptually he has identified the succeding stage to be the order of consciousness, or subject, which can take the previous order of consciousness as object. his diagrams graphically suggest what is happening here. previous models of developmental lines seem to have an arbitrary number of stages, which might be convenient for a particular purpose, but don't seem to me to get at what's really going on in development the way kegan's does.
ken has begun to talk about the number of perspectives one can take as a measure of development, a notion i think might be due to jane loevinger. this sounds alot like kegan's. by comparing them, it may be possible to establish which is more fundamental, in the sense, for example, that the cognitive line is fundamental to other lines of development. or it might point to a notion of development that is fundamental to both of these two. in either case we would achieve a more precise understanding of development, one that could more easily be astracted to create a new mathematics.
ken has been doing something like this, i would say, in putting together the aqal model, or framework, or map, of the kosmos. needless to say, the kosmos is a much, much more complex object than mathematicians have dealt with heretofore. aqal does not have the precision of a mathematical model. it's way too soon for that, since it's still being continually developed, but the concepts in which it is expressed already have an almost mathematical clarity.
some posters have been lamenting the absence of yotam schachter, who for some reason thinks he needs to get a bachelor's degree. this is my place to lament, i guess, for were he still here, i'm sure he would have a very interesting reply to my message.
sighingly,
ralph



03122007, 3:10 PM 

ralphweidner



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Re: "Integral Mathematics: An AQAL Approach"
first, an addendum to the previous post. i may have left the impression that probability waves invariably have a 25/50/25 shape, which is not at all the case. if it were, this model would be just as horribly mechanical as the previous three.
in fact, i think we can expect the shape of a probability wave to vary over time. i can imagine someone, in an initial test, having, say, a wave that is 25/50/25, with a green center of gravity, progressing in the course of three years, say, to 60% green, 30% teal, with the remainder at either end, and, in another three years, to 5% orange, 25% green, 60% teal and 10% turquoise, say. studying how these shapes change could tell us alot about development.
but before asking what integral math is, i should have asked what is math from an integral perspective. my initial feeling, without much thought, is that conventional math has pursued objectivity to its absolute extreme. can better examples of 'the view from nowhere' be found anywhere else? there is some faint recognition that mathematics is, after all, a human activity. we give credit to newton and leibniz for discovering calculus, going even so far as to retain a version of leibniz's notation. we name theorems after their discoverers or the first to prove them. but the math itself is to be completely shorn of any hint of subjectivity.
from an integral perspective i think we can conclude that conventional math has been primarily an orange affair that has become increasingly crippled and caught up in the performative contradiction of repressing anything hinting of subjectivity. of course, not always. i remember the mathematician michael spivak starting up publish or perish press in the 70's, and coming out with some voluminous volumes on differential geometry extolling the geometers who created this subject.
this is not to say that conventional math has been wrong, but terribly limited in its perspective. even so, it has accomplished alot, something we would want, by all means, to include as we attempt to expand its horizons.
i'm afraid this well be far from easy. it's very hard to break old habits. i can clearly see the value of looking at the four quadrants of a mathematician and at the four quadrivia of mathematics, but i'm not at all sure when i would be able to do it, as a matter of course. i feel more comfortable looking at other disciplines to see what they might contribute to the development of math, particularly those in zones 2 and 4.
ken has said, if i remember correctly, that there appear to be aspects of math that are eternal, that are involutionary givens. the first three models i looked at are all based on what i imagine would be one of those eternally givens: the circle.
let's examine this: was there a time when the circle did not exist? say, before ancient civilization? they certainly would not have understood it as the ancient greeks came to, so in that sense we can say that the circle did not exist for them. but certainly they were aware of the roundness of the sun, and a special time of the month when the moon reached full roundness.
if we go way back even to the bacteria of a billion or so years ago, i imagine they had some sense of circle, for example, of what was immediately around them. so i'm inclined to believe that the notion of a circle has always existed, but has been continually evolving from more primitive forms, and will go on evolving, but not in some arbitrary manner.
consequently, it behooves mathematics to begin coming to terms with the possibility that its particular notion of circle, however perfect it might appear, is not eternal, and begin taking into account its evolution, as well as that of all other mathematical entities, in order to pursue more effectively what is eternal.
this obviously requires more thought than i've given to it thus far,
ralph



03162007, 6:04 PM 

ben496



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Re: "Integral Mathematics: An AQAL Approach"
I would like to respond to the previous posts by Ralph to my Integral Mathematics article. I am quite pleased at the concise and accurate description that Ralph gave to my quasigroup model of consciousness. He nicely described my model as an algebraic way of describing the phenonmenon of returning to where you began, but somehow in a different place, and specifically in my application to a higher level of consciousness. However, I do believe that my article focuses upon a very different perspective of integral mathematics from what Ken has written about in his symbolic mathematics language, as described in "Integral Spirituality" and on the II website. Through my group theory and number theory examples I have tried to illustrate aspects of pure mathematics with applications to form a perspective of integral mathematics that is one of six perspectives I discuss in my article. In fact, I will say that in my in person meeting and telephone review call with Ken, I was inpressed by his openness to my viewpoint of integral mathematics, as I am coming from the perspective of being a pure mathematician and viewing mathematics as an art form, and this is a context that Ken graciously alllowed me to expand upon in my article and published it in AQAL. I welcome all other viewpoints of what Integral Mathematics is, and I will be happy to respond to any posts on Integral Mathematics in this forum.
Elliot Benjamin



03172007, 12:06 PM 

ralphweidner



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Re: "Integral Mathematics: An AQAL Approach"
hi elliot,
first, in going back to your article, i've noticed i unwittingly left off the 'bi' from your term 'biquasigroup'. i'm sure you put it there for a reason, so i'll edit that into my previous post.
i would enjoy being a member of an internet integral math group, if you decide to go ahead with that idea. it would be great to have yotam schachter as a member as well. he interned at ii last summer, has a special appreciation for the beauty of mathematics like yourself as well as remarkable facilityeven with ken's mathematics of perspectives.
i have the feeling we've only begun to scratch the surface. after reading your article, i got a hold of john stillwell's 'numbers and geometry' in order to learn more about number theory in conjunction with my specialty: an integral approach. it's a great introduction to higher mathematics that came out less than ten years ago. i hope to look at some of his other books as well, in particular, the 2nd edition of his 'history' of mathematics, which is obviously closely related to the evolution and development of mathematics. i guess it would be good to look at some of piaget's books as welland the book of gardner's you cite.
so much to do! simply to find out what others have done, but that's the integral approach, and it makes good sense: no point reinventing the wheel, or making a cart without them.
ralph



03182007, 4:53 PM 

ben496



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Re: "Integral Mathematics: An AQAL Approach"
Thanks Ralph for noticing the "biquasi" part of my biquasi group terminology. YesI did put in the biquasi term for the reason that this is all my own invention and the term "quasigroup" is already in use. Along these lines, I noticed that there is a significant editing error in the reference to my more extensive paper on this material for anyone interested in checking this out in more detail. In the last paragraph of my article, and in endnotes #22 and #23 the article mentioned should be my article cited in endnote #7, which is entitled "A Group Theoretical Mathematical Model Of Shifts Into Higher Levels Of Consciousness In Ken Wilber's Integral Theory" and it is available at www.integralscience.org And thanks for the numer thoery/geometry suggestion; I'll check this out some timethough it will probably be over the summer when I have the time to appreciate it. And by all means do suggest to the person you mentioned to chime right in here if he is interested in promoting some kind of integral mathematics network. When I met with Ken a few years ago, we discussed the possibility of getting together some kind of community gathering in Denver of people interested in forming an integral mathematics networkand I would guess that this would still be something II would have an interest in. So I think we have started the ball rolling hereand at this point we need more input from others. Lets see if there is enough interest in the II community for something to happen.
all the best,
Elliot



03182007, 5:34 PM 

ben496



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Re: "Integral Mathematics: An AQAL Approach"
iiadmin: By Elliot Benjamin
I would like to call attention to the significant error reference in my Integral Mathematics article for anyone interested in checking out my more extensive article concerning biquasi groups and consciousness level shifts. In the last paragraph of my article, and in Endnotes #22 and #23 the article cited should be the article listed in Endnote #7, which is entitled "On A Group Mathematical Model Of Shifts Into Higher Levels Of Consciousness In Ken Wilber's Integral Theory" and this is available at www.integralscience.org
Elliot Benjamin



03182007, 6:05 PM 

ralphweidner



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Re: "Integral Mathematics: An AQAL Approach"
hi elliot,
i'm not used to thinking or working with math in aqal terms, so i'm going to have work at it for awhile before i can expect some of my old habits to loosen their grip on me. i like your introducing various perspectives we can take on integral math, e.g. quadrants/quadrivia. even after one has recognized the importance of including most of these, if not all, in how we teach math and how we do it, it's so easy to leave some of this out.
i'm especially intrigued by what might constitute development of mathematics. i want to see what piaget and gardner said in this respect. stillwell is aware of the increasing complexity and abstraction to be found in mathematics with time, but i don't believe he's actually consulted any of the work of developmental psychologists.
on the other hand, i believe piaget only went as far as formop. i agree with your supposition that mathematicians go beyond formop and use visionlogic, for example, but no one has actually mapped this out, and i imagine it well take considerable effort to establish just what levels of cognition are present in the best of mathematics.
anyway, i'll try to spend some time in the coming months studying up on this. it sounds like you won't have much time until summer, and that's no doubt true for yotam, too, even assuming he's interested. he's in his last year at the university of chicago, majoring in both math and philosophy. the next generation! my generation is already past: i'm in my midsixties, and i imagine you're somewhere in between the two of us.
if we could do something this summer, that might get the ball rolling. in the meanwhile i'll keep you posted on whatever preparations i'm able to make.
ralph



03192007, 1:40 PM 

ben496



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Re: "Integral Mathematics: An AQAL Approach"
Hi Ralph,
I think we have a good plan. You might also want to check out the book "Where Mathematics Comes From" by George Lakoff and Rafael Nunez (it's referenced in my integral mathematics article).. They use the idea of metaphor to describe higher level abstract mathematical thinking, and it might give you some thoughts to ponder about mathematical thinking in its higher realms. And by the way, I am quite close to your own age, as I will be 57 next month. By all means do keep me informed about other ideas you develop concerning integral mathematics, as I think this discussion forum may be a good place to do that and perhaps others will eventually join us in this activity as well.
all the best,
Elliot



03192007, 11:06 PM 

ralphweidner



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Re: "Integral Mathematics: An AQAL Approach"
yes, that was one of a couple books you referenced that i want to check out.



04022007, 9:51 AM 

ralphweidner



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Re: "Integral Mathematics: An AQAL Approach"
elliott,
i'm going to post here my thoughts as they occur, knowing you'll probably be too busy to answer until at least summer.
i read mathematician mark ronan's book 'symmetry and the monster' last week. it's a short history of simple groups and their classification. reflecting on it and other integral matters as well, i'm feeling an ever increasing need to understand better ken's mathematics of perspectives. it's becoming more and more apparent to me that we really need to do this in order to clarify just what we're talking about.
of course, it can get really confusing. i can take a perspective of a perspective you took of a perspective i had taken of a perspective you had taken .... aqal provides a framework for introducing order. red can take a 1st perspective, amber a 2nd, orange a 3rd, green a 4th and 2nd tier, as presently constituted, a 5th. oh yeah! and ken can take a 7th, which is evidently what is needed to understand his integral math. implicit here, i think, is the notion of 'transcending and including': when we take perspectives of each other's perspectives, we probably aren't transcending and including each time, especially beyond 2nd tier, else we could join ken at higher atltitudes, i.e. red is healthy to the extent it has transcended and included a healthy magenta. to that extent we say it can take a 1st perspective. to the extent it has not, we would have to say it can't actually take a 1st perspective, and so on up the line, or stream. so we probably can't even take a truly 5th perspective yet.
this can be reversed, of course. if someone is talking intelligently about science, say, we can safely conclude she's able to take a 3rd perspective and, therefore, must be at least at an orange altitude.
i've said this before: i think zones 2 and 4 require a 4th perspective, i.e. green altitude.
your distinction between pure and applied math is one of perspective, i feel. i'm not sure of what other perspectives are involved nor their order, but at some point pure math takes, as you say, an LQ perspective, whereas applied math takes an RQ perspective.
interestingly, ronan, early on, makes a distinction between 'the real world' and 'the abstract world of mathematics'. that's very diplomatic and humble of him to imply that what he and colleagues do is not 'real', but, actually, it's just a matter of perspective. 'the real world' he is referring to is, of course, just the RQs, and at their lower levels, i.e. matter, the physical realm. 'the abstract world of math' is, in that respect, is to be found in the LQs.
he includes alot of historical and biographical material, but needless to say, he is not an historian nor a biographer by profession, although i still found it made for an interesting readit just didn't get very deep into what was really going on.
for that we need, imo, integral, developmental psychology, among other things.
one thing that intrigued me was his perspective, a very common one, that mathematics progresses by solving problems. i don't disagree with this, but it looks like a limited perspective to me that over emphasizes the RQs. what about the quest for beauty, for example? beyond his quest to solve a difficult problem, wasn't galois also after beauty? beauty that evidently had been denied him in 'the real world'?
unfortunately, when we see math as just the solving of problems, then that's what we tend to get. we could use a more expansive view!



04072007, 2:45 AM 

ralphweidner



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Re: "Integral Mathematics: An AQAL Approach"
just finished reading john derbyshire's 'unknown quantity, a real and imaginary history of algebra'. i love that subtitle! although i'm not sure what he had in mind.
he's very knowledgable, very up to date, with training in both mathematics and linguistics and work as a systems analystand he's very british: empirically oriented, witty.
he does proffer his own take on things, though. he's inclined to harold edwards' contructivist approach to mathematics, as opposed to the logical/set theoretic approach that dominated the last century, and that's the way he's written this book. he's somewhat dubious about 'intuitive' approaches, although he's pragmatic about it: if they work, that's fine with him.
he provides a simple, clear picture of how algebra has developed, perhaps too simple but good as an initial orientation.
he looks at it as a process of abstraction that develops at progressive levels (from specific to more and more variable) that encompass more and more objects: N, Z, Q, R, C, polynomials, ratios of polynomials, groups, rings, ...
how much of this is transformative and how much just translative? what galois did seems to me to have been truly transformative. it was a whole new way of looking at the solving of equations, clearly from a higher vantage point than previously, a higher order of consciousness, a larger perspective. was he taking, perhaps, a fifth perspective? that would be my guess, coming as it did after kant and about the same time as hegel and schelling.
to work this out, we would need to work out just what the equivalent of being able to take a 1st perspective is in mathematics; and a 2nd, 3rd and 4th.



04072007, 11:25 AM 

ben496



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Re: "Integral Mathematics: An AQAL Approach"
Hello Ralph,
To briefly respond to your previous two posts, I must say that my own personal interest in integral mathematics is not related to Ken's mathematics of perspectives, although ofcourse this is one of the perspectvies of integral mathematics that I acknowledged in my article. But what you mentoned about Galois I do find of more personal interest. Actually I see Galois theory as a beautiful mixture of pure and applied mathematics, or LQ and RQ mathematics. The application of the general insolvability of the quintic by radicals can be truly understood only after a very complicated and very aesthetic voyage into abstract algebra via group theory and field theory as formulated by Galois. I don't know how much his motivation went beyond the actual problem solving of proving the quintic has no general solutions in radicals, but what he achieved is most definitely of the highest order of mathematical thinking, in my opinion, and I have no doubt it would be on quite the high perspective scale in Ken's integral mathematics of perspectives. For another example of this beautiful inerplay of pure and applied mathematics, the highly abstract mathematics field of algebraic number theory, which is a combination of number theory and abstract algebra, was initially largely motivated by finding solutions of Diophantine equations, such as finding which integers can be written as the sums of two squares. And of course perhaps the supreme example is Andrew Wiles' proof in the 1990s of Fermat's Last Theorem, using various branches of abstract mathematics to prove that the Pythagorean Theorem cannot be generalized to higher exponents than 2.



04082007, 12:59 PM 

ralphweidner



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Re: "Integral Mathematics: An AQAL Approach"
hello elliot,
good to hear from you, even if you only have a moment to step in.
the difficulty for me with ken's mathematics, i believe, is that i simply don't have the requisite altitude, but i am drawn to wrestle with it from time to time simply because ken doesn't appear to yet have a handle on it himself. i wait for some precocious youth to step in and show us how surprisingly easy it is.
i believe it is actually a good, transformative practice to wrestle with it from time to time. my latest thoughts:
perspectives have a direction, going from the holon that is taking a perspective to its object. there is a view from, or through, and a view of, classified by quadrants (or octants), and quadrivia (or octavia?), respectively. as well as i can remember, ken focuses on individual holons taking perspectives. perhaps a collective holon can't really take a perspective? at any rate, my understanding is that a hyphen is inserted between the 1 or 3 (for an inside or outside view, respectively) and the p (for person) to distinguish taking a perspective from being taken in a perspectivefrom who is doing the viewing and who or what is being viewed. part of the confusion, i suspect, is that the latter are being included in tallying up the number of perspectives, and it shouldn't be. it's importance, of course, is that if we're going to be the object of a perspective, we would prefer, in general, that it be as a 1p, singular or plural, rather than a 3p.
btw, the 'integral spirituality' discussion group i belong to here in portland all agreed, when we got to zone 5 a few months ago, that it is confusing. what does it mean to take a 1p x 3p, or, even worse, a 3p x 1p x 3p? i asked if anyone had actually read anything from cognitive science, and no one had. obviously, this would help, so, in line with one of your recommended readingsas preparation, so to speaki've begun reading george lakoff's and mark johnson's book on the philosophy of the embodied mind (1999).
from past experience i have a deep trust in whatever ken says. he's coming through, again, in this case. what he's had to say in the excerpts and in 'integral spirituality' about this very new methodology clarifies so well what is going on here, in particular, the 3p x 1p x 3p. this is a valuable new perspective, but they're doing exactly what humanity has always done with any new perspective: they've absolutized it. their aim clearly is to conquer all previous perspectives (methodologies), not to work for an integral metamethodology or perspective, but for their own dominating methodology. what ken has pointed out with regard to varela holds here as well, even more so. they simply don't get what LQ methodologies are doing and have done. they include a mere shadow of them. they don't even seem to get what piaget was doing! so much for cognitive science, at least until it can become more integral.
this is disappointing. ken has been talking some time now how academia's love affair with deconstructive, zone 4 postmodernism has run its course and is coming to an end, the hope being that integral views will now be taken more seriously. but what if an absolutist, 'scientific' zone 5 succeeds in arresting the academic stage?
perhaps our difficulty in seeing this zone clearly, as in my discussion group, is due to our already being immersed in it! i recall, for example, how popular daniel goleman's book on social intelligence became when it came out last year. then there's the article by elliott ingersoll in the first issue of aqal journal, which i've commented on elsewhere, which seems to me to be coming from zone 5 without even being aware that it is doing so!? the message still is to live in the present moment. the hidden message, unfortunately, seems to be to first take a psychotropic drug.
algebraic number theory is something i need to learn more about, but it's going to have to wait i'm afraid. summer's just a season away, and there's so much else to do!



04092007, 8:56 AM 

ralphweidner



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Re: "Integral Mathematics: An AQAL Approach"
i actually had more to say about 'unknown quantity', which slipped my mind:
first, galois is the instigator, it appears to me, of the notion of structure in mathematics. didn't he introduce the notions of group structure and field structure, at least well beyond what anyone had done before him? and structuralism, which had become well established in math by the time i began graduate study in the mid60's, is a postmodern, green phenomenon. so this seems to enable us, as an initial estimation, to put galois' work at the level of green, indeed, the putative emergence of green in mathematics.
how far has math gone beyond green since then? my sense is not that far: that mathematicians are still doing what galois showed us to do, only with more sophistication.
what would a turquoise mathematics be like? it would have a much better grasp of development, in particular, imo. the lack of this is painfully evident to me in the books i've been reading by stillwell, ronan and derbyshire. the latter goes the farthest in his description of the development of algebra, but in comparison with developmental psychology, for example, it's horribly simplistic. if he were just aware of aqal, he would have realized that, for one thing, one needs to distinguish between the interior and exterior development of math. it's right there in what he says, but he doesn't see it. he frequently points out the lack of an adequate notation, and is amazed by what mathematician were nevertheless able to do. from an integral perspective, this is completely understandable, indeed, to be expected: there are interior (subjective) developments, which then become objectivized, exteriorized. conversely, exterior developments serve as a basis for further interior development. galois used the term 'group', but only in reference to possible manipulations of the coefficients of a polynomial: it was only completely exteriorized later on, so that, finally, we have the notion of an abstract group in its own right, which we can now make convenient use of, as an exterior object, in mathematics in any number of ways.
as a linguist derbyshire must be aware of development to a greater extent than other mathematicians, but he may be pretty much on his own at this point. incidentally, he wrote a short book on the riemann zeta function, as did harold edwards, before this book on algebra, which may be of interest.


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